Computational biomimetics of twisted plywood architectures ... · benign processing conditions using relatively simple raw materials but still manage to endow them with outstanding

Computational biomimetics of twisted plywood architectures in fibrous biological composites through chiral liquid crystal self-assembly

G. De Luca & A. D. Rey Department of Chemical Engineering, McGill University, Canada

Abstract

Despite being made of relatively simple materials, fibrous biological composites exhibit remarkable mechanical and physical properties [1]. These hierarchical materials frequently adopt a laminated architecture known as twisted plywood. In most cases this structure is monodomain (i.e. defect free), in which the fibrillar direction rotates around a single axis. However, not infrequently the twisted plywood architecture is found to be polydomain in which case there is multitude of local axis of rotation instead of a single one. In the latter case, the structure presents defects and it is therefore weakened. The origin of the twisted plywood structure is rather complex and has yet to be fully understood. Nevertheless it is strongly believed that liquid crystalline states are involved in its growth process. Indeed, numerous striking structural similarities between fibrous biological composites and these ordered fluid states have been observed and reported. The structure of liquid crystalline materials is known to be greatly dependent on the topology of their bounding surfaces. In this work, a mathematical model based on the Landau-de Gennes theory has been developed to investigate the role played by constraining surfaces in the structural development of a composite material that undergoes a liquid crystalline state during the early stages of its growth. The goal of this study is to investigate the role played by constraining surface on these materials. The numerical simulations qualitatively confirm the hypothesis of Neville [2, 3], according to which the presence of a constraining surface produces a mechanically effective monodomain structure whereas its absence leads to a weakened polydomain organization. In addition to these results, this approach highlights the role played by modelling in the study of tissue morphogenesis. Keywords: biological composites, twisted plywood, chiral nematic liquid crystals.

Design and Nature II, M. W. Collins & C. A. Brebbia (Editors)© 2004 WIT Press, www.witpress.com, ISBN 1-85312-721-3

1 Introduction

Unlike engineers, Nature assembles a number of composite materials under benign processing conditions using relatively simple raw materials but still manage to endow them with outstanding mechanical properties, Vincent [1]. One essential reason for that is their unique hierarchical design. Developing a fundamental understanding of the design and processing practices of Nature is of a major interest for advances in applied material science.

Fibrous composites are widespread in Nature; they are found in animal skeletal systems and in plants. Each and every of these composite materials is made of rather different building blocks but however they often present a common architecture. This organization is known as twisted or helicoidal plywood and has been widely reported and illustrated in the literature. An entire textbook by Neville [3] is devoted to this subject.

A distinguishing feature of the twisted plywood architecture found in biocomposites is that it resembles the characteristic organization of typical chiral nematic liquid crystals, also known as cholesteric liquid crystals. In these mesophases, molecules are laying on a series of equidistant pseudo-layers. Each layer is characterized by a common molecular orientation, however from one layer to the next this direction is slightly rotated which gives rise to a periodic structure. The pitch, denoted by p0, corresponds to the distance required by the average molecular orientation to rotate by 2π radians along the helical axis of the structure.

The current understanding is that in the early stage of their morphogenesis, natural composites, undergo a liquid crystalline phase that allow us to self-assemble in the twisted plywood fashion before subsequently solidify [4, 5]. Liquid crystallinity is therefore thought to play a fundamental role in the structural development of these materials.

The structures of liquid crystalline mesophases are recognized to be greatly affected by bounding surfaces. Surface irregularities can, for example, lead to distortions and defects in the structure of the mesophases. Their presence in fibrous biological composites is of a great importance as they can modify the mechanical functionalities of the material.

When a chiral nematic mesophase forms by bulk nucleation and growth without any constraints from bounding surfaces, a polydomain structure comporting many local helical axis is usually expected to form. The incompatibilities in orientation of the neighbouring helices results in topological defects. However, when the chiral nematic mesophase is allowed to develop in the presence of constraining surfaces, for example by means of a directed front propagation, its structure is expected to be a defect-free monodomain. Whether, the structure is monodomain or polydomain, is a key issue which governs the mechanical performance of the material, and thus establishing the conditions that lead to a perfect defect-free monodomain is important. Neville [2] concluded that the defect-free monodomain twisted plywood architecture arises when the material forms on a constraining surface. This constraining layer forces the chiral


474 Design and Nature II

nematic mesophase to growth with a unique helical axis which avoids the nucleation of topological defects.

There is a recognized need for theories and computational studies to complement experimental work directed to the understanding of structure formation in biological materials. In this work we use a well-established theoretical framework of liquid crystals in order to investigate the different textures formation modes of chiral nematic mesophases and to improve the current understanding of structural organization in fibrous biological materials.

2 Theory and governing equations

Liquid crystalline mesophases are characterized by a long-range orientational order that is usually mathematically described by means of a tensor order parameter, Q. In its most general format, this second-order symmetric and traceless tensor reads: 1

3 3( ) ( ,S P= − + −δQ nn mm ll) (1) where the following restrictions apply: T ;=Q Q (2) tr( 0;=Q) (3) 1

2 1;S− ≤ ≤ (4) 3 3

2 2 ;P− ≤ ≤ (5) 1;⋅ = ⋅ = ⋅ =n n m m l l (6) .+ + =nn mm ll δ (7)

The unit vectors n, m and l form an orthogonal director triad which

characterizes the average preferred molecular orientation of the phase. The unit vector n is known as the uniaxial director, and m and l as biaxial directors. The quantities S and P, known respectively as uniaxial and biaxial scalar order parameters, are measure of the average molecular alignment. The uniaxial scalar order parameter gives the degree of alignment along the uniaxial director n while the biaxial scalar order parameter gives the degree of alignment along the first biaxial director m. The tensor order parameter Q characterizes the microstructure of the mesophase by combining information about the molecular orientation and alignment. The correspondence between phase and alignment are: isotropic (S=0, P=0), uniaxial (S≠0, P=0) and biaxial (S≠0, P≠0).


Design and Nature II 475

The dynamical governing equation used to describe the structure evolution of the chiral nematic mesophase is derived from the classical Landau-de Gennes theory [6] by using variational principles associated with the free energy functional. In its dimensionless form, this non-linear partial differential equation reads: t−∂ = +Q H G, (8) where: [s]

3(1 ) ( ) ( ) ,U U U= − − ⋅ + :H Q Q Q Q Q Q (9)

0 0 0 0

2 2 [s] 2h h p p( ) 8 ( )( )( ) 16 ( ) .ξ ξ ξ ξ= − + π × + πG Q Q Q∇ ∇ (10)

The tensors H and G represent the homogeneous and gradient

contributions to the dynamics of Q; U is the nematic potential driving the phase-transition between the isotropic and the ordered phase; ξ is a correlation length associated with the variation of the scalar order parameter S; h0 corresponds to the external length scale or thickness of the sample; p0 is the pitch of the chiral nematic mesophase at equilibrium; ξ/h0 is a ratio that controls the balance between transitional and gradient effects while ξ/p0 rules the equilibrium chirality of the mesophase (In the limit where this later ratio tends to zero, the material describes an ordinary achiral nematic liquid crystalline mesophase); [s] indicates that only the symmetric and traceless part of the tensor is retained. The governing eqn (8) was solved for the five independent components of the tensor order parameter. For all the different simulations presented here, the mesophase was assumed to be initially in an isotropic state. Physically, this initial disorder of the material corresponds to a low concentration of axisymmetric fibrous molecules in the matrix of the forming biological composite. The ordering of the composite is triggered by an increase of the concentration in fibrous molecules beyond a critical value. In the model, this information is carried out through the parameter U that can be seen, for the system considered, as proportional to concentration. The parametric values used for the different simulations reported in this article are as follow: U = 6, ξ/p0 = 0.06 and ξ/h0 = 0.02. The set of values lead to representative results that best capture the main objective of this report.

3 Results and discussions

3.1 Formation of the polydomain twisted plywood structure

We first consider the formation of the polydomain twisted plywood structure that emerges when the chiral nematic material is allowed to grow unconstrained by any bounding surfaces. This structure forms by nucleation, growth and impingement of spherulites. The impingement of the spherulites leads to the



formation of grain boundaries which produces defects when intersecting. This defect nucleation process is known as Kibble mechanism [7]. In order to simulate this polydomain formation process, we solve governing eqn (8) using the following periodic boundary conditions: x xx 0 x 1

(x 0) (x 1), ,= =

= = = ∂ = ∂Q Q Q Q (11)

y yy 0 y 1

(y 0) (y 1), .= =

= = = ∂ = ∂Q Q Q Q (12)

With this type of boundary conditions, an image of the system is

periodically repeated in space. Figure 1a-d shows the formation of a polydomain twisted plywood structure. The ordering process starts at t=0+ as the dimensionless parameter U suddenly increases from 2.5 to 6, at which the homogeneous isotropic state becomes unstable. During this phase-ordering process orientation (n, m, l) and alignment (S, P) evolve in order to generate a new ordered stable phase. Due to some initial fluctuation of order, the chiral nematic mesophase emerges faster in some region of the computational domain. Since those domains are embedded in an unstable isotropic phase, they are not subjected to any constraints from the boundaries. In these regions, the material is hence free to grow in a spherulitic way; the material is twisting into every radial direction from the center of the domains. The emerging structure is analogous to the double-twist configuration. The appearance of such organization in the material bulk is due to the fact that the system has a lower free energy compared to that of the single twist configuration. As time evolves, the chiral nematic spherulites grow until they finally impinge their closest neighbours. When this event occurs, the spherical domains slightly modify their peripheral orientational order and become polygonal and the whole system reaches a new stable structure. This steady-state microstructure is a polygonal network formed by grain boundaries. Within these polygonal domains, the nematic director n describes a half-pitch going from the center of the domain to its boundaries. In the center of the domains, the director n lays normal to the computational x-y plane. Figure 1a-c show a time series of the out-of-plane component |nz| of the director field using grayscale plots. The field goes from white to black as |nz| goes from 1 to 0. Figure 1a represents an early stage of the process when chiral nematic spherulites emerge from the unstable isotropic bulk. Black rings appear sporadically indicating the circular nature of the twist. Figure 1b represents the impingement stage at which neighbouring domains are coalescing. Figure 1c illustrates the steady-state structure of the material which consists of a network of polygonal domains. The walls of those domains, which appear in black in the plot, correspond to in-plane orientations of the directors. A defect is found at each intersection between three domain walls.

Figure 2 represents a grayscale plot of the uniaxial scalar order parameter S at steady state. The field goes from white to black as S goes from high to low values. The black dots, corresponding to defects, are located at the triple junctions of the grain boundaries.



Figure 1a: Out-of-plane component |nz| of the director field at dimensionless

time t=2e-4. The field goes from white to black as |nz| goes from 1 to 0.

Figure 1b: Out-of-plane component |nz| of the director field at dimensionless time t=3.2e-3.



Figure 1c: Out-of-plane component |nz| of the director field at steady state,

t=8.9e-3.

Figure 2: Grayscale plot of the uniaxial order parameter S at steady state. The field goes from white to black as S decreases.



The polygonal network given by the steady-state solution of our Landau-de Gennes based model is hence a stable defect lattice.

3.2 Formation of the monodomain twisted plywood structure

We now consider the formation of the chiral nematic texture under the influence of bounding surfaces. As mentioned previously, when a chiral liquid crystalline material is allowed to grow on a smooth constraining surface, the structure that is expected to appear is a monodomain twisted plywood. The unique helical axis of the structure is normal to the supporting surface. Undoubtedly, the relief of this surface plays a key role in the conformation of the material in the bulk. If in industry, the topography of a surface can be controlled to some extent by rubbing techniques and chemical treatments, the situation is a little bit different in Nature as such preparation can be a priori absent. It is hence interesting to examine how the monodomain structure can still be achieved in biological fibrous materials where the surface is not necessarily smooth.

In order to account for such situation we ideally assume a grooved support much like in the spirit of Berreman [8]. Along this wavy surface, the material is considered to be in a stable equilibrium chiral nematic state.

Figure 3a: Out-of-plane component |nz| of the director field at steady-state.

The nematic director n is assume to be tangentially oriented to the

grooves while the equilibrium values of the uniaxial and biaxial scalar order parameters are given by the solutions of the following autonomous system of coupled differential equations:



( ) ( )( ) ( )

0

0

22 2 2 3 22 1 1 2 19 9 3 3 3 pt

22 3 22 2 2 13 3 9 3 pt

4 ( ),

12 ( ).

S P S P S S S U S S P

P S P PS P P U P S P

ξ

ξ

∂ = − − + − + − − π −∂ = − − − + − + π −

(13)

Figure 3a shows a grayscale plot of the out-of plane component of

director field |nz| at steady-state. It appears clearly from this picture that the distortion induced by the supporting surface is progressively absorbed as progressing in the bulk of the material. The characteristic length of this perturbation is comparable to the periodicity of the grooves and therefore to the length of the dimensionless chiral nematic pitch p0. Finally, figure 3b presents the case of a more irregular surface topography in which the groove have greater amplitude and are non-symmetric. Once again, we can notice that the few first layers of the material undergo orientational readjustments that progressively lead to a monodomain twisted plywood structure in the bulk of the material. The thickness of the transition region is, as in the previous case presented, proportional to the periodicity of the grooves. This last example is hence reinforcing the idea that support imperfections, as they may arise in Nature, are probably smooth-out by a reorientation mechanism.

Figure 3b: Out-of-plane component |nz| of the director field at steady-state.

4 Conclusions

A Landau-de Gennes based model of chiral nematic liquid crystalline material has been used to investigate the role played by constraining surfaces in the



structural development of fibrous biological composites. Numerical results obtained with this model agree with the hypothesis postulated by Neville [2] according to which in the absence of constraining surfaces, composites undergoing a chiral nematic liquid crystalline phase during their morphogenesis are likely to display a polydomain structure. In order to display the more mechanically effective monodomain twisted plywood architecture, the material must be allowed to grow on a supporting surface that constrains the system to have a single helical axis. Disturbances induced by the relief of this supporting surface are naturally quenched over a distance commensurate with wave length of the distortion. Past this distance, the material bulk exhibits the undistorted monodomain planar twisted plywood architecture. These results qualitatively agree experimental observations [2, 3]. This work was supported by a grant from the Donors of The Petroleum Research Fund (PRF) administrated by the American Chemical Society.

References

[1] Vincent, J.F.V., Structural biomaterials, Princeton University Press, 1991. [2] Neville, A.C., The need for a constraining layer in the formation of

monodomain helicoids in a wide range of biological structures, Tissue & Cell, 20(1), pp. 133-143, 1988.

[3] Neville, A.C., Biology of Fibrous Composites, Cambridge University Press, 1993.

[4] Bouligand, Y., Liquid crystals and theirs analogues in biological systems, Solid State Physics Supp. 14, Academic Press, pp. 259-294, 1978.

[5] Giraud-Guille, M.M., Twisted liquid crystalline supramolecular arrangements in morphogenesis, International Review of Cytology, 166, pp. 59-101, 1996.

[6] De Gennes, P.G., Prost, J., The Physics of liquid Crystals 2nd edition, Oxford, 1993.

[7] Kibble, T.W.B., Cosmology of cosmic domains and strings, Journal of physics A, 9(8), pp. 1387-1398, 1976.

[8] Berreman, D.W., Alignment of liquid crystals by grooved surfaces, Molecular crystals and liquid crystals, 23(3-4), pp. 215-231, 1973.



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Computational biomimetics of twisted plywood architectures ... · benign processing conditions using relatively simple raw materials but still manage to endow them with outstanding